1. Field of the Invention
This invention relates to an intensity distribution simulating method, and particularly to an intensity distribution simulating method for use in a lithographic process for manufacturing semiconductor devices.
2. Description of the Related Art
Photolithography has been mainly utilized in a lithographic process which is one of the processes for manufacturing semiconductor devices. The recent development of photolithography has enabled mass-production of semiconductor devices having the minimum line width (resolution) below 0.5 .mu.m because a projection lens of a demagnification projection aligner (a reducing projection type of exposure device) can be designed to have a higher numerical aperture (NA). The increase of the numerical aperture (NA) enables improvement in resolution, but it increasingly causes reduction in depth of focus. Therefore, in the case where a finer pattern having a line width below 0.35 .mu.m is required to be formed, it has become increasingly impossible that the mass-production of semiconductor devices is stably performed by merely increasing the NA (numerical aperture) of the projection lens.
In order to overcome this circumstance, much attention has been paid to a so-called oblique illumination technique in which the resolution characteristics can be improved by optimizing an illumination system. For example, Japanese Laid-open Patent Application No. Sho-61-91662 which is titled "PROJECTION DEVICE" proposes a method which is required to perform the above oblique illumination technique for a demagnification projection aligner. In this proposed method, a specific diaphragm is disposed just behind an integrator (fly-eye lens) which is an optical element for improving in-plane homogeneity of illumination, thereby changing the shape of an effective light source as described later. The integrator is an optical element which is formed by bundling the same type of several tens unit (simple, or single) lenses each of which has the rectangular and slender shape. The respective unit lenses are individually focalized to form the effective light source.
An extra-high pressure mercury lamp is used as an original light source for illuminating patterns (reticle patterns) on a mask in a general demagnification projection aligner. The extra-high pressure mercury lamp emits light in a relatively broad range, so that the light emitted from the original light source does not have a high coherence. However, if the integrator is used in this system, the coherence of lights which are emitted from the ultra-high pressure mercury lamp and focused by the respective unit lenses of the integrator becomes higher, so that these lights can function as independent point-sources of light. Accordingly, it can be obtained as an imaging characteristic by this effect that the illumination is not affected by the shape of the original light source (ultra-high pressure mercury lamp), but affected by the shape of only the point-source group of the integrator. Therefore, the group of the point-sources which constitute the integrator is called an "effective light source".
Generally, in order to resolve an image pattern, of all diffracted lights, zero-order diffracted light and +primary diffracted light or -primary diffracted light are required to be collected. However, if a diaphragm 6 shown in FIG. 7A is used in a projection device as shown in FIG. 7B, a diffraction angle .theta. of diffracted light from a reticle (mask) 7 becomes larger as a pattern is finer, so that the diffracted light is not incident to a projection lens system 8 and thus no pattern is resolved. On the other hand, if a ring-shaped diaphragm 6' shown in FIG. 7C is used in a projection device as shown in FIG. 7D, light emitted from the integrator is incident to the reticle (mask) 7 only in an oblique direction, so that any one of +primary diffracted light and -primary diffracted light is incident to the projection lens system 8. Therefore, a finer pattern can be resolved. In addition to the circular zonal illumination technique as described above, a four-point illumination technique of illuminating light at four points is also used to particularly improve the resolution characteristics of longitudinal and lateral patterns. In these cases, the optimum diaphragm is selected in accordance with each reticle pattern to broaden the manufacturing margin of the semiconductor devices.
As described above, at the recent stage where the requirement in the market approaches to the functional limitation of the photolithography, it has been increasingly required to select the optimum exposure method in accordance with each reticle pattern to broaden the process margin as much as possible. In order to satisfy this requirement, a simulation method for determining the optimum exposure system becomes more important. That is, a light intensity distribution simulation which is matched with various kinds of oblique illumination techniques as described above has been increasingly required to keep sufficient process margin in accordance with various reticle patterns. However, in the oblique illumination technique, abnormal deformation of patterns may occur due to a proximity effect of the exposure system although the resolution and the depth of focus are improved. In order to avoid such an adverse effect due to the proximity effect, an accurate consideration based on the intensity distribution simulation has been increasingly indispensable for the manufacturing of the semiconductor devices.
Next, a general light intensity simulation will be briefly described.
Hopkins theory is most well known as one for the light intensity simulation method. According to the Hopkins theory, in the case of a partially coherent illumination system, the intensity distribution I(x,y) of a reticle pattern over an image plane is calculated on the basis of the Fourier transform of the object (reticle pattern or mask pattern) transmission function F(x',y') and of the inverse Fourier transform of the coherent transmission cross-coefficient (hereinafter referred to as "TCC") function. The coherent transmission cross-coefficient represents the degree of variation of the intensity distribution due to coherence, and it is calculated on the basis of the pupil function of the projection lens system and the light intensity distribution function S of the effective light source. Accordingly, according to the light intensity simulation using the Hopkins theory as described above, the light intensity distribution I(x,y) represented by the equation (1) is calculated according to the following procedures (i) to (iv) (in the following equations, (x',y') represents a coordinate on an object (reticle or mask), (x,y) represents a coordinate on an image plane (photoresist face), and the coordinate system of these is shown by FIG. 10: ##EQU1## Procedure (i) calculates the Fourier transform of a reticle pattern as where F(f,g) of equation (2) is the spatial spectrum of the object transmission function F(x',y'); ##EQU2## Procedure (ii) calculates the TCC F(f,g;f',g') of the optical system; procedure (iii) calculates the double integral of equation (3) for the results of procedures (i) and (ii); ##EQU3## and procedure (iv) takes the inverse of Fourier transform of the function of procedure (iii) to obtain I(x,y).
The treatment of the optical system on the basis of the coherent transmission cross-coefficient TCC requires the numerical integration to be carried several times (Hopkins theory requires quadruple integration), so that this theory is theoretically excellent, but it is not practical. In view of the disadvantage of the Hopkins' method as described above, the M.Yeung's method (Proceedings of the Kodak Microelectronics seminar INTERFACE '85, (1986) PP115-126) is used because it is suitable for computer's calculation.
According to the Hopkins theory as described above, the characteristics of the optical system, that is, the characteristic of the effective light source and the characteristic of the projection lens system are collectively treated with only the coherent transmission cross-coefficient TCC. On the other hand, according to the Yeung's method, these characteristics are individually and independently treated with. This method is described in Japanese Laid-open Patent Application No. Hei-3-216658, for example. In this method, the effective light source of the integrator is divided into a finite number of point sources, and a projection image due to the whole effective light source of the integrator is obtained by superposing respective projection images due to the respective point sources over one another. In this method, the following assumption is introduced. That is, light emitted from each point source is treated as perfect coherent light, and a projection image due to each point source is incoherent to projection images due to the other point sources. Accordingly, if a projection image due to each point source is calculated and then all the projection images due to the respective point sources are summed up (superposed), the projection image due to the integrator (effective light source) could be obtained.
Here, the Yeung's method as described above will be briefly described with reference to FIG. 8.
First, an effective light source is divided into point sources as shown in FIG. 8. In this case, the number of the point sources is represented by n. Here, the light emitted from an i-th point source is represented by an unit vector Si, and the components (direction cosines) of the unit vector Si on the coordinate system (x',y') of the object plane (reticle plane) are represented by pi and qi. In this case, the amplitude of the light emitted from the i-th point source incident on the object plane is given by a function of Ai and the direction cosines pi and qi: EQU Ai*exp{j(2.pi./.lambda.)(pix'+qiy')},
where .lambda. represents the wavelength of the light in the medium and Ai represents a complex number characterizing the intensity and phase of the light of the i-th point source. Accordingly, the amplitude of transmission light due to the i-th point source through the reticle (hereinafter referred to as "the reticle transmission light amplitude of the i-th point source") is represented by the product of the above function and the object (reticle) transmission function F(x',y'): EQU Ai*F(x',y')exp{j(2.pi./.lambda.)(pix'+qiy')}.
Here, let us consider a projection image corresponding to a reticle pattern due to the i-th point source (coherent light source). Now, the coherent transfer function of the optical system is represented by K(x--x',y--y'). This function is the amplitude of the light arriving at the point (x,y) on the image plane due to an unit point source at the point (x',y') on the object plane. The amplitude distribution Ui of light over the image plane due to the lights emitted from all parts of the object (reticle) plane is obtained by the integral calculus of the reticle transmission light amplitude of the i-th point source and the coherent transfer function K of the optical system. The intensity distribution of the projection image of the reticle pattern due to the i-th point source is equal to the second power of the amplitude distribution Ui, and the light intensity distribution I(x,y) of the whole effective light sources (n pieces) is represented as follows: ##EQU4## That is, I(x,y)=.SIGMA..vertline.Ai.vertline..sup.2..vertline.amplitude value on the image plane due to the i-th point source.vertline..sup.2
Here, if all the point sources of n are of equal intensity, then .vertline.Ai.vertline..sup.2 =1/n for all i=1, 2, . . . , n, where n represents the total number of the point sources.
The equation (4) is still not in the most convenient form for the computer calculations. Using the convolution theorem in the Fourier transformation theory, the above equation may be rewritten as follows: ##EQU5## where F'(f,g) is the Fourier transform of the object (reticle) transmission (distribution) F(x',y') and given by the equation (2), and K'(f,g) is the Fourier transform of the coherent transfer function K(x--x',y--y') of the optical system. Therefore, the intensity distribution I(x,y) is represented by the sum of the squared absolute value of the inverse Fourier transform of the product between the Fourier transform K'(f,g) of the coherent transfer function K(x--x',y--y') and the Fourier transform F'(f-pi/.lambda.,g-qi/.lambda.) of the reticle transmission distribution F(x',y'). The function K(f,g) is calculated on the basis of the pupil function P. Therefore, the intensity distribution I(x,y) is calculated according to the following computer 's calculation procedure:
(1) calculate the Fourier transform of the reticle pattern;
(2) calculate the Fourier transform K' of the coherent transfer function K with the pupil function P;
(3) fix the direction cosines pi and qi for each point source, and then calculate the product of the functions F' and K' for each point source;
(4) take the Fourier transform of the result of (3), and then add the second power of the absolute value of the obtained Fourier transform to the whole intensity distribution I(x,y); and
(5) repeat the calculations of (3) and (4) for all the point sources.
This simulation method has a smaller number of integration steps than the method using the Hopkins theory, and thus it is more suitable for the computer's calculations.
In the conventional intensity distribution simulation methods as described above, the effective light source which constitutes the integrator of the reticle illumination system is regarded as an assembly of the point sources. Each point source is allocated to each unit lens 2 (cell) of the integrator, and these unit lenses are assembled into the integrator 1 as shown in FIG. 9A. The actual intensity distribution of light just after emitted from each cell (unit lens) of the integrator 1 is shown in FIG. 9B.
As is apparent from FIG. 9B, the light emitted from each cell 2 (corresponding to a point source) has an intensity distribution with its peak at the center thereof, and thus the total light intensity of the light from the integrator 1 has a discrete intensity distribution as shown in FIG. 9B. However, according to the conventional intensity distribution simulation methods as described above, the intensity distribution of the light emitted from the integrator (point sources) is treated as being uniform (fixed), that is, the intensity of the lights from the respective point sources are regarded as being uniform (fixed). If each cell of the integrator is fine and the number of the cells is above several tens, the number and position of the point sources (cells) have no effect on the imaging characteristics of the exposure system. Actually, an integrator whose cells are designed to be as fine as possible is used in the demagnification projection aligner for the manufacturing of the semiconductor devices to improve uniformity of illumination. Accordingly, in this case, it is unnecessary to take the layout of the integrator 1 into consideration.
Accordingly, in the intensity distribution simulation using the Hopkins theory, the effective light source in the integrator is generally treated as having an uniform intensity distribution over the effective area when TCC is calculated. Furthermore, even in a special case, consideration is paid to only the macro-level difference in light intensity (the difference in light intensity between the center and the periphery of the integrator). Furthermore, in the intensity distribution simulation using the Yeung's method, no consideration is paid to the layout of the integrator when each effective light source is divided into a finite number of point sources.
However, in the case where the oblique illumination such as the circular zonal illumination or four-point illumination is performed, the area of the aperture of the diaphragm becomes small and the number of effective cells of the integrator which are actually used for exposure also becomes small. Therefore, in this case, it has become more problematic to treat the effective light source as a light source having uniform intensity. That is, the present situation becomes serious to such an extent that the simulation result varies in accordance with the intensity distribution in the aperture of the diaphragm for the oblique illumination and the position of the point sources. Therefore, it is difficult to accurately simulate the projection image of an actual reticle pattern unless the intensity distribution of each cell of the integrator is individually considered.